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A064237 Numbers n such that n! + 1 is divisible by a square. 6
4, 5, 7, 12, 23, 229 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

229 is another term because 613^2 divides 229!+1. See A115091 for primes whose square divides m!+1 for some m. An examination of the factorizations of m!+1 for m<=100 found no additional squares. - T. D. Noe, Mar 01 2006

562 is also a term because 562!+1 is divisible by 563^2. - Vladeta Jovovic, Mar 30 2004

Comment from Francois BRUNAULT, Nov 23 2008: A web search reveals that for 1 <= n <= 228 there are 82 values of n for which n! + 1 has not been completely factored (the smallest is n=103), so showing that 229 and 562 are indeed the next two terms will be a huge task. I checked that n!+1 is not divisible by p^2 for n <= 1000 and prime p < 10^8.

It is very likely that 229 and 562 are the next two terms, but this has not yet been proved. - Nov 29 2008

Contains A007540(n)-1 for all n.  That sequence is conjectured to be infinite. - Robert Israel, Jul 04 2016

This sequence includes A146968 (solutions of Brocard's problem). -  Salvador Cerdá, Mar 08 2016

I may be wrong, but perhaps it is not yet proved that there are no terms between 23 and 228? - N. J. A. Sloane, Jul 24 2017

LINKS

Table of n, a(n) for n=1..6.

Hisanori Mishima, Factorizations of m!+1

EXAMPLE

4 is in the sequence because 4! + 1 = 5^2.

5 is in the sequence because 5! + 1 = 11^2.

6 is not in the sequence because 6! + 1 = 721

7 is in the sequence because 7! + 1 = 71^2.

12 is in the sequence because 12! + 1 = 13^2 * 2834329.

23 is a term because 23!+1 =47^2*79*148139754736864591.

229 and 562 are terms because

229!+1 = 613^2 * 38669 * 1685231 * 3011917759 * (417-digit composite)

562!+1 = 563^2 * 64467346976659839517037 * (1275-digit composite)

MAPLE

remove(t -> numtheory:-issqrfree, [$1..50]); # Robert Israel, Jul 04 2016

MATHEMATICA

Flatten[Position[MoebiusMu[Range[30]!+1], 0]]; (* T. D. Noe, Mar 01 2006, Nov 21 2008 *)

PROG

(PARI) lista(nn) = for(n=1, nn, if(!issquarefree(n!+1), print1(n, ", "))); \\ Altug Alkan, Mar 08 2016

CROSSREFS

Cf. A007540 (Wilson primes), A115091, A146968, A038507, A085692.

Sequence in context: A112918 A138920 A048224 * A139373 A293757 A275273

Adjacent sequences:  A064234 A064235 A064236 * A064238 A064239 A064240

KEYWORD

nonn,more

AUTHOR

Vladeta Jovovic, Sep 22 2001

EXTENSIONS

Example corrected by T. D. Noe, Nov 26 2008

STATUS

approved

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Last modified November 22 05:36 EST 2017. Contains 295076 sequences.